DSpace Collection:http://hdl.handle.net/2440/10872014-09-22T14:14:20Z2014-09-22T14:14:20ZThe Guillemin-Sternberg conjecture for noncompact groups and spacesHochs, P.Landsman, N.http://hdl.handle.net/2440/853602014-09-18T22:58:27Z2007-12-31T13:30:00ZTitle: The Guillemin-Sternberg conjecture for noncompact groups and spaces
Author: Hochs, P.; Landsman, N.
Abstract: The Guillemin–Sternberg conjecture states that “quantisation commutes with reduction” in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved for compact Lie groups G acting on compact symplectic manifolds, and, largely due to the use of Spinc Dirac operator techniques, has reached a high degree of perfection under these compactness assumptions. In this paper we formulate an appropriate Guillemin–Sternberg conjecture in the general case, under the main assumptions that the Lie group action is proper and cocompact. This formulation is motivated by our interpretation of the “quantisation commuates with reduction” phenomenon as a special case of the functoriality of quantisation, and uses equivariant K-homology and the K-theory of the group C*-algebra C*(G) in a crucial way. For example, the equivariant index – which in the compact case takes values in the representation ring R(G) – is replaced by the analytic assembly map – which takes values in K0(C*(G)) – familiar from the Baum–Connes conjecture in noncommutative geometry. Under the usual freeness assumption on the action, we prove our conjecture for all Lie groups G having a discrete normal subgroup Γ with compact quotient G/Γ, but we believe it is valid for all unimodular Lie groups.2007-12-31T13:30:00ZA dynamical systems approach to simulating macroscale spatial dynamics in multiple dimensionsRoberts, A.MacKenzie, T.Bunder, J.http://hdl.handle.net/2440/852472014-09-11T23:39:32Z2013-12-31T13:30:00ZTitle: A dynamical systems approach to simulating macroscale spatial dynamics in multiple dimensions
Author: Roberts, A.; MacKenzie, T.; Bunder, J.
Abstract: Developments in dynamical systems theory provide new support for the macroscale modelling of pdes and other microscale systems such as lattice Boltzmann, Monte Carlo or molecular dynamics simulators. By systematically resolving subgrid microscale dynamics the dynamical systems approach constructs accurate closures of macroscale discretisations of the microscale system. Here we specifically explore reaction–diffusion problems in two spatial dimensions as a prototype of generic systems in multiple dimensions. Our approach unifies into one the discrete modelling of systems governed by known pdes and the ‘equation-free’ macroscale modelling of microscale simulators efficiently executing only on small patches of the spatial domain. Centre manifold theory ensures that a closed model exists on the macroscale grid, is emergent, and is systematically approximated. Dividing space into either overlapping finite elements or spatially separated small patches, the specially crafted inter-element/patch coupling also ensures that the constructed discretisations are consistent with the microscale system/pde to as high an order as desired. Computer algebra handles the considerable algebraic details, as seen in the specific application to the Ginzburg–Landau pde. However, higher-order models in multiple dimensions require a mixed numerical and algebraic approach that is also developed. The modelling here may be straightforwardly adapted to a wide class of reaction–diffusion pdes and lattice equations in multiple space dimensions. When applied to patches of microscopic simulations our coupling conditions promise efficient macroscale simulation.2013-12-31T13:30:00ZMechanics of fullerene-carbon nanotube bundle oscillatorsThamwattana, N.Baowan, D.Cox, B.Hill, J.http://hdl.handle.net/2440/851162014-09-09T03:24:06Z2007-12-31T13:30:00ZTitle: Mechanics of fullerene-carbon nanotube bundle oscillators
Author: Thamwattana, N.; Baowan, D.; Cox, B.; Hill, J.
Abstract: Carbon nanostructures, such as nanotubes and fullerenes, are of considerableinterest to a wide range of research communities. Owing to their unique properties, a numberof applications involving these structures have been proposed, one of which is the so-callednano-scaled oscillators, for which the resultant oscillatory frequencies are in the gigahertzrange. The phenomena of the gigahertz oscillations have also led to the possible creation offuture devices, such as ultra-fast optical filters and ultra-sensitive nano-antennae. In thispaper we present recent developments in this area and review the authors recent work,where we investigate the mechanics of a new type of gigahertz oscillator comprising afullerene C60 oscillating within the centre of a uniform concentric ring or bundle of carbonnanotubes. Using the Lennard-Jones potential and the continuum approach, for which carbonatoms are assumed to be uniformly distributed across the surface of a molecule, we providethe underlying mechanisms of these nano-scaled oscillators.2007-12-31T13:30:00ZHighest density difference region estimation with application to flow cytometric dataDuong, T.Koch, I.Wand, M.P.http://hdl.handle.net/2440/850972014-09-09T02:47:36Z2008-12-31T13:30:00ZTitle: Highest density difference region estimation with application to flow cytometric data
Author: Duong, T.; Koch, I.; Wand, M.P.
Abstract: Motivated by the needs of scientists using flow cytometry, we study the problem of estimating the region where two multivariate samples differ in density. We call this problem highest density difference region estimation and recognise it as a two-sample analogue of highest density region or excess set estimation. Flow cytometry samples are typically in the order of 10 000 and 100 000 and with dimension ranging from about 3 to 20. The industry standard for the problem being studied is called Frequency Difference Gating, due to Roederer and Hardy (2001). After couching the problem in a formal statistical framework we devise an alternative estimator that draws upon recent statistical developments such as patient rule induction methods. Improved performance is illustrated in simulations. While motivated by flow cytometry, the methodology is suitable for general multivariate random samples where density difference regions are of interest.2008-12-31T13:30:00Z